By Edward J. Beltrami
This new version of arithmetic for Dynamic covers instruments similar to linearization, suggestions innovations, using Liapunov services, and optimum keep an eye on. each one bankruptcy contains workouts, lots of which extend at the fabric within the textual content
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Additional resources for Mathematics for Dynamic Modeling
A system in which energy is lost is called dissipative. In the next chapter, the situation E < 0 will be treated more carefully using the notion of a Liapunov function. It will be seen that the orbits of the damped pendulum equation tend to a rest point as a result of a decrease in total energy. The system runs itself down because of energy dissipation due to damping. Let us close by giving a last example. Some familiarity with elemen tary electric circuit theory is required to understand how the present model comes about.
11. 11 in which xx is position, and x2 is velocity. 48 Stable and Unstable Motion, II and to the right in this quadrant, meaning that movement is clockwise. s. tells us that any orbit spirals down to zero. 2 Stable Equilibria, II The discussion that follows is a bit more technical. 3. The reader may wish to skip this section on a first reading. 1 in some region Ω. 13) v } for each component g, of the vector g. 13. Assume, for simplicity, that the eigenvalues Xi oi A are real, distinct, and negative.
6). It is necessary to resolve these forces into compo nents which are, at each instant, along the actual direction of motion, namely along the tangent to the circular arc of radius /. 6 Force diagram for the inverted pendulum. Feedback 55 hand's lateral motion exerts a force on the mass in the opposite direc tion. An acceleration of the hand to the right moves the pivot left. By Newton's law, the restoring force is therefore v = — mw. The total force on the mass is then ms = mg sin 0 — kYs — mw cos 0.
Mathematics for Dynamic Modeling by Edward J. Beltrami